B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

64 ANALYSIS AND TRANSMISSION OF SIGNALS
Fi g
ure 3.2
Change in the
Fourier spectrum
when the period
To in Fig. 3.1 is
doubled.
Envelope
/ t G(f)
0
(a)
0
/ Ern;cloJ<
. · ./ l_G(f)
•••••, , , , 1 rr I JI I j 11 n r", , , , ...
(b)
D" l
Note, however, that the relative shape of the envelope remains the same [proportional to G(f)
in Eq. (3.3)]. In the limit as To -+ oo,Jo -+ 0 and D n -+ 0. This means that the spectrum is
so dense that the spectral components are spaced at zero (infinitesimal) interval. At the same
time, the amplitude of each component is zero (infinitesimal). We have nothing of everything,
yet we have something! This sounds like Alice in Wonderland, but as we shall see, these are
the classic characteristics of a very familiar phenomenon.*
Substitution of Eq. (3.5) in Eq. (3.1) yields
(3.6)
As To -+ oo,Jo = 1 /To becomes infinitesimal (Jo -+ 0). Because of this, we shall replace Jo
by a more appropriate notation, D.f . In terms of this new notation, Eq. (3.2b) becomes
1
D.f = - To
and Eq. (3.6) becomes
00
gTo (t) = L [G(nD.f)D.f] e (j 2n:nD. f) t
n=-oo
(3.7a)
Equation (3.7a) shows that gTo (t) can be expressed as a sum of everlasting exponentials of
frequencies 0, ±1::i,,f, ±2D.f, ±3D.f, • .. (the Fourier series). The amount of the component
of frequency nD.f is [G(nD.f)D.f]. In the limit as To -+ oo, D.f -+ 0 and gT0 (t) -+ g(t).
Therefore,
00
g(t) = lim gr 0
(t) = lim L G(nD.f)e U 2n:nD.f)t
D.f (3.7b)
To ➔
oo
D.f'
O ➔ n=-oo
The sum on the right-hand side of Eq. (3.7b) can be viewed as the area under the function
G(f)d2rr ft, as shown in Fig. 3.3. Therefore,
* You may consider this as an irrefutable proof of the proposition that 0% ownership of everything is better than
100% ownership of nothing!